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Evans, 1965; Linlor, 1980; Ambach and Denoth, 1980; Colbeck, 1982; Tiuri, 1982]. For dry snow, the real part of relative permittivity varies from 1.2 to 2.8, depending on the snow density. The imaginary part varies from 10-4 to 10- 2 . For wet snow, the real part of relative permittivity varies from 2 to 6, and the imaginary part varies from 10-3 to 1, depending on temperature, wetness, and frequency. Thus it is desirable that the theory of scattering by dense media should reduce to a good mixture formula at very low frequency. A study of the geometry and grain structure of snoW has been done by Colbeck [1972, 1979, 1982].
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Vegetation
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Vegetation consists of leaves and stalks embedded in air, such as alfalfa, sorghum, corn, soy beans, wheat, and so on. The fractional volume occupied by leaves and stalks per unit volume of vegetation including the air space is between 0.1% and 1%. The particles in vegetation are nonspherical in shape with large aspect ratios. Leaves have the shape of thin disks, and stalks assume the form of long slender cylinders. They also have preferred orientation distribution. Scattering by nonspherical particles can give strong depolarization return. Depending on the type of vegetation, the thickness of leaves are of the order of 0.1 mm to 1 mm and the surface area can vary from 1 cm 2 to 103 cm 2 . Stalk diameters vary from 1 cm to 5 cm, and the length can vary from 5 cm to 100 cm. For geometry and inclination characteristics of leaves and stalks in sorghum, one can refer to Havelka [1971]. Very little theoretical and experimental study has been done on the variation of the permittivity of vegetation as a function of moisture and frequency. One permittivity model that has been used is the following [de Loor, 1968; Fung and Ulaby, 1978]:
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(4.3.4) M Vw = ---,----::--::-:-----,---,---,(4.3.5) dw/d s + M (1 - dw/d s ) where A is the free space wavelength in centimeters, Vw is the volume filling
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4 CHARACTERISTICS OF DISCRETE SCATTERERS AND ROUGH SURFACES
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factor of the dispersed granules in leaves, M is the moisture contents by weight, ds is the density of the solid material, and dw is the density of water.
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The atmosphere is a dispersed medium with randomly distributed aerosols and hydrometeors in the air. As electromagnetic waves propagate in the atmosphere, the wave energy is absorbed and scattered by these liquid or solid particles. It has great effects in the applications of communication and remote sensing using electromagnetic waves. The pertinent physical properties are size distribution, concentration, chemical composition, shape, and orientation distribution. Aerosols are particulate matter suspended in the atmosphere, examples include smog, smoke, haze, clouds, fog, and fine soil particles. Their sizes are generally under 1 /-lm in radius. Hydrometeors are water particles in solid or liquid form in the atmosphere. Some examples are mist, rain, freezing rain, ice pellets, snow, hail, ocean spray, clouds, and fog, whose sizes are generally 1 /-lm or more in radius. Aerosols and hydrometeors are not of same size, they are usually characterized by a broad range of sizes. The electromagnetic properties, such as attenuation and depolarization, of a volume of aerosols or hydrometeors depend strongly on their size distributions, composition, and shapes. Rain is one of the most important hydrometeors. The size distribution of rain droplets depends on the precipitation rate (rain rate) p, which is normally expressed in millimeters per hour (mm/hr) [Chu and Hogg, 1968]. Rainfall is made up of roughly spherical water droplets. Raindrops can be characterized in terms of the diameter 2a, with a falling terminal velocity v(a). Let n(p, a) da be the number of rain droplets per unit volume having radius between a and a+da at the precipitation rate p. An empirical rain drop size distribution given by Marshall and Palmer [1948] is of the exponential form
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where no = 8 x 106 m- 4 , a = 8200p-O.2I m- I , a is in meters, and p is in millimeters per hour. Note that even though there are more particles with smaller radii, these small particles have a relatively small effect on wave propagation and scattering. To determine the effect of rain on wave propagation, we also need to know the falling terminal velocities of raindrops, which are dependent on the drop radius. It has been shown that over the diameter range 1 through 4